Now before you go any further, put the date Saturday 12th October in your diary! That is the date of the next Maths conf which is in Peterborough. If you have not yet been to a Mathsconf, I cannot recommend it enough! I have now been to 3 mathsconf and I walk away from each one inspired and content in the knowledge that it is come of the best maths CPD that I receive.
The day started off like the other Maths confs I have been to with a quick intro to the day and some words from the key people that run the event. It was a pleasure to watch Mark receive an award from Macmillan for all of the support that he has given to them. It is brilliant the money that gets raised at mathsconf for Macmillan via the sales of raffle tickets and the tuck shop. Macmillan is certainly a charity that is close to my heart and so it is lovely to go to a conference and be see the support that they receive.
Andrew from AQA then led us through a ‘guess the year’ game using past exam questions.
The below question is from 1988 (O-Level)
The question below is from 2018 (GCSE)
The question below is from 1962 (O-Level)
The question below is from 1994 (GCSE)
The question below is from 1940 (school cert)
It was interesting to see how questions have changed throughout the years and the contrast between the way in which the questions are worded and asked.
After this there was ‘speed dating’. You are asked to come with a favourite resource to share with someone else and given just 90 seconds to explain to your resource to someone, they then have 90 seconds to explain their resource to you. I think in total we had about 5 dates. I shared my idea of splitting my board into 3 segments, red, amber and green, and then placing a question in each segment (red being easy up to green being hard). I then give the students about 3-5 minutes to have a go at the questions on whiteboards. I then go through each question and get the students to mark their answers and hold up their boards. From here, I give them a relevant starting point. Its a nice way to differentiate tasks.
A particular resource that I was shown during these dates was a pirate trigonometry activity. This has been used with students in the classroom on several occasions and I was told the students enjoyed this activity and that currently not one of the groups had yet completed the task within an hour. If you want to check this out, simply google tes pirate game and it will appear.
After this, we went into the sessions that we had selected.
Session 1 – Making Mathematicians by Kate Milnes (twitter handle @katban70)
This was a brilliant session, Kate shared so many wonderful ideas.
She started with the concept that mathematicians start with a scenario ask themselves questions and pursue a line of inquiry – is this what we make our students do?
Kate then took us through an example to encourage our students to conjecture:
- Write down 3 different digits
- write largest possible number
- write smallest possible number
- Subtract smallest from largest
- Write this result forwards
- Write it backwards
- Ask yourself a question
If you follow these steps correctly you end up with 1089. Kate has done this with students and the most common question was does it wok with 4 digits? Students then worked through these steps again with 4 digits. The outcome for 4 digits is 10890. Kate then encourages her students to conjecture. Some of them conjectured that 5 digits would be 108900. They then worked through this and realise that it does not come to 108900. Apparently, some students became deflated at this point but Kate wanted to teach them that this is an important part of mathematics and instead of becoming deflated they should be asking ‘why didn’t it work’ and then continue. This then led to a generalisation:
This is such a great way to get students thinking about why things happen and encouraging them to prove it. Proof plays such an important part in Mathematics and yet it is something that we maybe don’t spend a lot of time on. Whilst on an A-Level course it was a common consensus that students seemed to find proof a hard part of the course. The person running the course then asked a very apt question ‘how many of you use proof in KS3 and KS4?’ We said that we tend to touch on it at GCSE as it is on the syllabus but we don’t put a lot of emphasis on it. The course leader simply turned and said ‘they probably find it hard as it is generally the first time they have really come across it in any depth. It is important to implement proof in KS3/KS4 where you can’. This really stuck with me and so it was brilliant to hear Kate talking about encouraging students to conjecture and to generalise.
Kate then talked about variation theory and using it as a tool to encourage students to conjecture:
By getting the students to use the table above, it encourages them to think about how a simple change will have an affect on the outcome and to conjecture about this new outcome. What a great way to implement practise and conjecture into the classroom.
One other resource to encourage students to pursue a line of inquiry is to use goal free problems. There is a website dedicated to these problems: http://goalfreeproblems.blogspot.com/ or you can create them yourself by snipping exam questions. The idea behind this is that you give students information but you don’t give them an actual question. They create the question for themselves and glean what they can from what they have in front of them. I know that a colleague has used these in their class and commented that the amount of work the students did was incredible.
I can’t wait to use these resources in my lessons. Thank you Kate for an excellent session!
Session 2 – Plan lessons not learning by Gary Lamb (twitter handle @garyl82)
This was again a brilliant session. I loved that Gary was not afraid to be controversial. He warned us at the start of the session that he was going to include statements that might go against the norm but it was great for Gary to do this.
statement 1: Every child can achieve at least a grade 5 in GCSE maths by the end of year 11.
Gary said that for a department this is a great ambition (and not unrealistic).
Statement 2: Get through the content and do the revision at the end
A comment that Gary has heard said before. His view, teach it right the first time.
This led onto quality of instruction:
The sweet spot combines teacher-directed instruction (in most lesson) and inquiry (in some lessons).
The SOL that Gary uses is approx 80% teacher directed. Gary’s view: we should not be afraid to teach kids.
Statement 3: Lesson starters are not an effective use of lesson time.
Most lesson starters are pre-requisites, could these questions not be set as a pre-task before the lesson? Gary did recognise that some teachers use starters as a way to settle students and to buy some time to get sorted (this may be essential if you have to move between classrooms when teaching) but said that we should not be afraid to question something just because we have always done it.
Gary then went through what some of his lesson slides look like:
I have used a very similar style to the one above ever since reading Craig Barton’s ‘how I wish I had taught Maths’. He talked about direct instruction and going through an example and then getting the students to do their own question. I remember when I read this it made me feel as if I had permission to teach again. This is what I loved about what Gary said, he made me feel OK about the fact that I do spend a lot of time going through examples and getting my students to do very similar questions. A few examples like this may be followed by:
I really liked the three different sections here. I thought it was a great way to get the students thinking about the basic skill in different ways. Gary said that not all students will complete all of these questions but that is ok, there should be something there for each student. Those who have really got it might get all of them complete which gives Gary time to get to those students struggling with this skill and only just getting through the do stage.
You might be thinking there are not a lot of questions here but Gary showed us some results from some research:
The high Masser students completed 9 questions during the lesson and the Lo massers completed 3. 4 weeks later when they were tested on the same skill there was only a 1% difference in retention.
After the basics of expanding these brackets you may wish to extend students onto expanding brackets with negative numbers. Instead of just giving them questions on directed numbers you could use more thinking based tasks for this such as venn diagram problems (visit mathsvenns for lots of these)
Other activities that could be used to assess understanding is ‘here is the answer, what is the question’ or ‘which is the odd one out?’.
Gary’s session was certainly thought provoking and I thought a brilliant example of what lessons can look like. I will certainly be using these ideas in my own lessons; thank you Gary.
Session 3 – The inverse department by Jason Steele and Luke Modiri (twitter handles @steelemaths and @modirimaths)
This session was all about collaborative working as to reduce work load. Lots of us have families that require our attention and so do not want to spend all of our time at work. However, we don’t want to sacrifice good teaching as our students deserve our best when we are at work; this was a method to try and help staff be at their best.
The department that Jason and Luke work in use their department meeting time in a different way. The HOD and 2nd decide upon a topic, at the department meeting the department discuss what they expect students to learn. This is done in the form of a mind map. Take for example solving equations with year 7, what would we expect our students to learn? Solving equations with one variable, solving equations with negative and fractional answers, solving equations with a variable on both sides of the equal sign, forming and solving basic equations. From this, the department then share ideas and knowledge on this topic e.g. what would this knowledge look like for our higher attainers and our lower attainers? An example of this is below:
The department agree upon a common approach of teaching this area. The approach used must be one that will work all the way through mathematics and not one that will break down as the topic becomes harder. I know of some schools that agree with this and also use common approaches. The idea is that if students end up having different teachers teach them, the methods don’t change dependant upon whom they have teaching them. It is supposed to help with retention.
Once the department have done this an individual teacher goes away and finds examples, questions, resources and creates a series of lessons. The lessons are shared out amongst the teachers meaning that each teacher does not end up creating more than approx 18 lessons a year. The idea is not that these lessons are adhered to rigidly or used as a supply lesson but that each teacher adapts them accordingly for the needs of their students, The department that use this method all agreed to have this as a target on their appraisal.The final stage is to ensure consistency and feedback to the teachers, this is achieved by the department agreeing upon certain requirements and standards to adhere to e.g. a lesson presentation should not be more than 10 slides etc
This session was again thought provoking and certainly presented an interesting approach to lesson planning.
Session 4 – AQA GCSE 2019 – How was it for you? by Andrew Taylor (twitter handle @AQAmaths)
The school that I work for currently use AQA for our GCSE examinations and so it always great to be able to go along to a session ran by AQA.
Andrew shared some interesting ideas behind the purpose of the multiple choice questions:
- At start of paper to be accessible
- Reliable and valid
- To test misconceptions
- To support other approaches
Andrew used this question above as an example of how multiple choice can be used to help students. He argued that if the question was simply asked without the multiple choice element some students may find it more difficult.
Andrew then discussed what is meant by accessible:
It was refreshing to hear this from an exam board. As Andrew said in the session, accessible does not mean easier but it should mean that most students have an idea of where to being with a question even if they can’t work through the whole question.
Next Andrew discussed how layout of questions can be used to help students access questions:
The table here has been used so that a lot of description could be taken out of the question. It is quite clear that this question involves mass, volume and density and it is quite clear what elements need finding.
Andrew then went on to talk about getting the demand right:
- The right question in the right place for the right purpose
- All assessment objectives tested
- Appropriate for tier
- Discriminate based on mathematical ability only
The question above was asked in a foundation paper and was question 8. This is a problem solving question. It is the belief of AQA that A03 questions should be in the foundation papers and that problem solving should not be the preserve of the latter end of the paper. This question however provides students with a route through the question; the information is given in a way to help students access this question.
Below is another example of an accessible question found on a higher tier paper:
This is question 27 on the paper so very close to the end. This question is deemed as accessible as a lot of candidates should be able to recognise that the radius of the larger circle is 12. This question should not only be started by the grade 9 students, but grade 7/8 students should also be able to make a start on this question, although not expected to pick up the full 5 marks. Remember, accessible does not mean easy, it means that students should be able to make a start.
In conclusion, to make questions clear:
- clean layout used
- Useable diagrams
- Care taken with language
- Ambiguity avoided
- Layout used to support students
A very interesting session, thank you Andrew for sharing these insights.
Session 5 – Variation theory – More than minimally different questions by Chris Mcgrane (twitter handle @chrismcgrane84)
Chris suggested that instead of looking at Maths as topics, we should look at it as themes:
- freedom and constraint
- Invariance in the midst of change
- extending and restricting meaning
- doing and undoing
These themes permeate throughout mathematics.
Variation theory is not a new idea. It looks as what is the same and what is different, what has changes and what has stayed the same.
One concept of variation is perceptual variation:
The idea of this bar model above is to get the students to look at it differently. The final 2 questions encourage the students to stop looking at the bar as a whole but look at it in a different way, It is a variation on how we usually present bar models.
Chirs then went through 3 different ways in which we can introduce prime numbers to our students. The point being there were 3 procedures for the same outcome. Chris then challenged us by asking how many different ways can we prove the identity (sinx)^2 + (cosx)^2 = 1. Personally, I know of one way, by using a right angle triangle and applying Pythagoras. Chris then argued that we should know 3 different ways into this or we should be questioning if our understanding is good enough to really be teaching this.
What Chris said next married up with what Kate had discussed earlier, that variation theory goes beyond pattern spotting, it should encourage our students to ask the question why and lead to generalisations. Variation theory should be looking at relationships.
After the final session we all headed back to the main hall and the prizes for the raffle and treasure hunt were handed out and the Macmillan total announced. The final thing of every mathsconf is then to send a postcard to to yourself. This is a CPD postcard with what you are going to try in your lessons after being at mathsconf, On mine I wrote that I want to have implemented some goal free tasks into my lessons. These postcards are then posted a few months down the line. I only hope that when mine arrives I have trialled goal free tasks.
Once again, a massive thank you to everyone that makes mathsconf happen. It truly is one of the best days of maths CPD that I receive, Don’t forget the next maths conf is in Peterborough on Saturday 12th October. GET YOUR TICKET!